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原文摘要

We study interactive proofs for arithmetic circuits over the complex numbers $\mathbb{C}$ in which both the prover and the verifier compute approximately (unlike muggles, wizards can use magic, i.e., floating-point numbers). The goal is to prove that a claimed output $y$ is close to the exact output $C(x)$: for thresholds $0<Δ_0<Δ$, the honest prover is accepted whenever $|y-C(x)|\leΔ_0$, and every prover is rejected with high probability whenever $|y-C(x)|>Δ$. Existing proof systems for real-valued data verify rounded finite-field computations. Bitan et al.~[EUROCRYPT~2026] gave an approximate sum-check protocol. We extend the protocol of Goldwasser, Kalai, and Rothblum (GKR) [STOC~2008] to public-coin interactive proofs for logspace-uniform layered circuits over $\mathbb{C}$ of size $S$, depth $L$, and $n$ inputs, under public bounds on gate values and wiring coefficients. At constant soundness error, the verifier runs in time $(n+1)\mathrm{poly}(L,\log S,b)$, the communication is $\mathrm{poly}(L,\log S,b)$, and the prover runs in time $\mathrm{poly}(S,b)$, where $b$ is the bit precision. To the best of our knowledge, these are the first doubly efficient interactive proofs in which both parties compute approximately. GKR reduces two claims about a layer to one on a random line, which leaves the unit circle, where our error bounds hold. We instead label gates by complex roots of unity and combine the two claims with a random root of unity as coefficient. We choose the error thresholds backward from the input layer, so that a large output error survives the verifier's rounding at every step. The honest prover needs only $O(S \log S)$ complex operations. We also obtain zero-knowledge proofs from one-way functions, one-round arguments from private information retrieval, and interactive PCPs whose proof length is polynomial in the witness length.